An analytical model for azimuthal thermoacoustic modes in an annular chamber fed by an annular plenum

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Bauerheim, Michaël and Parmentier,

Jean-François and Salas, Pablo and Nicoud, Franck and Poinsot, Thierry

An analytical model for azimuthal thermoacoustic modes in an

annular chamber fed by an annular plenum. (2014) Combustion and

Flame, vol. 161 (n° 5). pp. 1374-1389. ISSN 0010-2180

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An analytical model for azimuthal thermoacoustic modes in an annular

chamber fed by an annular plenum

Michaël Bauerheim

a,⇑

_{, Jean-François Parmentier}

a

_{, Pablo Salas}

b

_{, Franck Nicoud}

c

_{, Thierry Poinsot}

d

a_{CERFACS, CFD team, 42 Av Coriolis, 31057 Toulouse, France}

b_{INRIA Bordeaux – Sud Ouest, HiePACS Project, Joint INRIA-CERFACS Lab. on High Performance Computing, 33405 Talence, France} c_{Université Montpellier 2, I3M UMR CNRS 5149, Place E. Bataillon, 34095 Montpellier, France}

d_{IMF Toulouse, INP de Toulouse and CNRS, 31400 Toulouse, France}

Keywords: Azimuthal modes Analytical Combustion instabilities Coupling

a b s t r a c t

This study describes an analytical method for computing azimuthal modes due to flame/acoustics cou-pling in annular combustors. It is based on a quasi-one-dimensional zero-Mach-number formulation where N burners are connected to an upstream annular plenum and a downstream chamber. Flames are assumed to be compact and are modeled using identical flame transfer function for all burners, char-acterized by an amplitude and a phase shift. Manipulation of the corresponding acoustic equations leads to a simple methodology called ANR (annular network reduction). It makes it possible to retain only the useful information related to the azimuthal modes of the annular cavities. It yields a simple dispersion relation that can be solved numerically and makes it possible to construct coupling factors between the different cavities of the combustor. A fully analytical resolution can be performed in specific situa-tions where coupling factors are small (weak coupling). A bifurcation appears at high coupling factors, leading to a frequency lock-in of the two annular cavities (strong coupling). This tool is applied to an aca-demic case where four burners connect an annular plenum to a chamber. For this configuration, analyt-ical results are compared with a full three-dimensional Helmholtz solver to validate the analytanalyt-ical model in both weak and strong coupling regimes. Results show that this simple analytical tool can predict modes in annular combustors and investigate strategies for controlling them.

1. Introduction

Describing the unstable acoustic modes that appear in annular gas turbine combustion chambers and finding methods to control them are the topic of multiple present research activities[1–9]. The complexity of these phenomena and the difficulty of perform-ing simple laboratory-scale experiments explain why progress in this field has been slow for a long time since. Recently, the devel-opment of smaller annular chambers in laboratories has opened the path to investigating flow fields[10,11], ignition[12], flame re-sponse to acoustics[13], and azimuthal instabilities in these con-figurations[4,14,15]. At the same time, theoretical and numerical approaches have progressed in three directions: (1) full 3D LES of annular chambers has been developed [16,17], (2) 3D acoustic tools have been adapted to annular chambers [18–21], and (3) analytical approaches have been proposed to avoid the costs of 3D formulations and to investigate the stability and control of modes at low cost[5,22,23]. This last class of approaches is espe-cially interesting for elucidating mechanisms (such as transverse

forcing effect[23], symmetry breaking[5], and mode nature[24]) because they can provide explicit solutions for the frequency and the growth rate of all modes. The difficulty in these methods is in constructing a model that can be handled by simple analytical approaches while retaining most of the important physical phe-nomena and geometrical specificities of annular chambers.

One interesting issue in studies of instabilities in annular cham-bers is classifying them. For example, standing and turning modes

[1,24] are both observed[1,5,17,25], but predicting which mode

type will appear in practice and whether they can be studied and controlled with the same method remains difficult[23]. Similarly, most large-scale annular chambers exhibit multiple acoustic modes in the frequency range of interest (typically 10–30 acoustic modes can be identified in a large-scale industrial chamber between 0 and 300 Hz), and classifying them into categories is the first step in controlling them. These categories are typically ‘‘longitudinal vs azimuthal modes’’ or ‘‘modes involving only a part of the chamber (decoupled modes) vs modes involving the whole system (coupled modes)’’[1,3,20,21]. Knowing that a given unsta-ble mode is controlled only by a certain part of the combustor is an obvious asset for any control strategy. In the case of combustors including an annular plenum, burners, and an annular chamber, ⇑ Corresponding author.

such a classification is useful, for example, in understanding how azimuthal modes in the plenum and in the chamber (which have a different radius and sound speed and therefore different frequen-cies) can interact or live independently. For example, FEM simula-tions of a real industrial gas turbine [20] produce numerous complex modes that involve several cavities (plenum, burners, and chamber) at the same time. Unfortunately, determining whether certain parts of a chamber can be ‘decoupled’ from the rest of the chamber is a task for which there is no clear strategy. ‘Decoupling’ factors have been derived for longitudinal modes in academic burners where all modes are longitudinal[26]. Extending these approaches to annular systems requires first deriving analyt-ical solutions that can isolate the effect of parameters on the modes structure. This is one of the objectives of this paper.

Noiray et al.[5]and Ghirardo and Juniper[23]have proposed analytical analysis of complex nonlinear mechanisms, but only in simple configurations where the combustor is modeled with an annular rig alone. Parmentier et al.[22]have derived an analytical method called ATACAMAC (for analytical tool to analyze and con-trol azimuthal modes in annular combustors) for a more realistic configuration called BC (N burners + chamber) (Fig. 1 left). By describing acoustic wave propagation and flame action in a net-work of ducts representing the BC configuration and introducing a reduction method for the overall system corresponding to wave propagation in this network, they were able to predict the frequen-cies and growth rate of azimuthal and longitudinal modes, to iden-tify their nature, and to predict their response to passive control methods such as symmetry breaking. Stow and Dowling[27]also investigated BC configurations with low-order models and focused on limit cycles by introducing more complex flame models (by adding nonlinearities and uniform spread of convection times).

However, BC geometries did not correspond exactly to real annular chambers where the N burners are connected not only downstream to the combustion chamber but also upstream to the plenum that feeds them. PBC configurations (plenum + N burn-ers + chamber) (Fig. 1right) have been proved[1,20]to correctly reproduce the behavior of complex industrial annular combustors

(Fig. 2). Evesque and Polifke[8]studied PBC configurations using

both FEM simulations and low-order models. Pankiewitz and Satt-elmayer[21]also investigated this case using time-domain simula-tions with an axial mean flow. Both the linear and nonlinear flames regimes are studied by introducing saturations in the flame trans-fer functions. They pointed out that the time delay of the FTF plays a crucial role in predicting the frequency as well as the nature of azimuthal modes in such a configuration.

Thus, no full analytical resolution of the frequency and nature of azimuthal modes has been achieved in PBC configurations. The present paper extends the analytical methodology of Schuller et al.[26]for longitudinal tubes and of Parmentier et al.[22]for BC configurations to a PBC configuration with N burners in order to highlight key parameters involved in the coupling mechanisms. In most network approaches to combustion instabilities, a very large matrix is built to describe the acoustics of the system

[8,26,28]. Here, we introduce a significantly simpler methodology

called ANR (annular network reduction) that makes it possible to reduce the size of the acoustic problem in an annular system to a simple 4-by-4 matrix containing all information of the combustor resonant modes. This method makes it possible to obtain explicit dispersion relations for PBC configurations and to exhibit the exact forms of the coupling parameters for azimuthal modes between the plenum and the burners, on one hand, and between the burn-ers and the chamber, on the other hand.

The paper is organized as follows: Section2describes the prin-ciple of the ANR (annular network reduction) methodology and the submodels that account for active flames. The decomposition of the network into H-shaped connectors and azimuthal propagators makes it possible to build an explicit dispersion relation giving the frequency, growth rate, and structure of all modes. In Section3, thermoacoustic regimes (from fully decoupled to strongly coupled) are defined, depending on the analytical coupling parameters con-ducted in Section2. Finally, this analytical model is validated using the model annular chamber described in Section4with simplistic shapes to construct coupling factors and study azimuthal modes for a case where a plenum is connected to a chamber by four sim-ilar burners (N ¼ 4). First the weakly coupled regime (Section5) and then the strongly coupled regime (Section6) are investigated. Nomenclature

a

¼ zf ;i=Li normalized abscissa for the flame location in the

burners b¼c0Lp

c0

uLc tuning parameter



pand



cwavenumber perturbation in the plenum and chamber Ci;kkth coupling parameter of the ith sector

F ¼q_q00c0 uc0uð1 þ nie

jxsi_{Þ flame parameter}

x

angular frequency

q

0

uand

q

0 mean density in the unburnt and burnt gases

s

i time delay of the FTF of the ith flame

s

0 p¼2c 0 u pLp and

s

0 c¼2c 0

pLc period of the unperturbed pth azimuthal

mode of the plenum and chamber h angle in the annular cavities c0

uand c0mean sound speed in the unburnt and burnt gases

kuand k wavenumber in the unburnt and burnt gases

Lc¼

p

Rcand Lp¼

p

Rp half perimeter of the chamber or plenum

Li length of the ith burner

N number of burners

ni interaction index of the FTF of the ith flame

p order of the azimuthal mode p0 _{pressure fluctuations}

Rcand Rp radii of the annular chamber or plenum

Sc; Sp, and Si cross sections of the chamber, plenum, and ith

burner

u0 _{azimuthal velocity fluctuations (along x)}

w0 _{axial velocity fluctuations (along z)}

x azimuthal abscissa in the chamber or plenum corre-sponding to xc¼ Rchor xp¼ Rph

z longitudinal abscissa in the burners ANR annular network reduction

ATACAMAC analytical tool to analyze and control azimuthal modes in annular combustors

BC burners + chamber configuration (Fig. 1left)

BCp burner + chamber mode of order p (the annular plenum is perfectly decoupled from the system)

FDCp fully decoupled chamber mode of order p FDPp fully decoupled plenum mode of order p FTF flame transfer function

LES large eddy simulation

PBC plenum + burners + chamber configuration (Fig. 1right) PBp plenum + burner mode of order p (the annular chamber

is perfectly decoupled from the system) SCp strongly coupled mode of order p WCCp weakly coupled chamber mode of order p WCPp weakly coupled plenum mode of order p

The bifurcation[29]from weakly to strongly coupled situations is triggered by increasing the flame interaction index controlling the flame response to the acoustic flow. Results show that ATACAMAC makes it possible to predict azimuthal turning and standing modes in a PBC configuration and performs as well as a 3D Helmholtz sol-ver for all three regimes, while simple annular rigs[5,23]or BC models[22,27]are not able to capture the bifurcation from weakly to strongly coupled regimes.

2. A network model for PCB (plenum + burners + chamber) configurations

2.1. Model description

The model is based on a network view of the annular chamber fed by burners connected to an annular plenum (Fig. 3). This model is limited to situations where pressure fluctuations depend on the angle h (or x) but not on z in the chamber and the plenum (they de-pend on the coordinate z only in the N burners). This case can be observed in combustors terminated in choked nozzles that behave almost like a rigid wall (i.e., w0_{¼ 0 under the}

low-upstream-Mach-number assumption[30]). Note, however, that this restriction pre-vents the present model from representing academic combustors where the combustion chamber is open to the atmosphere

[4,14,15]and from predicting ‘‘mixed modes’’ appearing at higher

frequencies in configurations terminated by choked nozzles[20].

Since the chamber inlet is also close to a velocity node, modes that have no variation along z can develop in the chamber, as shown by recent LES[31]. Radial modes (where p0 _{depends on r)}

are neglected because they often occur at high frequency. Gas Fig. 1.Configurations for studying unstable modes in annular chambers.

Fig. 2.FEM simulations performed by Campa et al.[20]on a complex industrial gas turbine (only six sectors are displayed) and its PBC configuration model. Two kinds of eigenmodes are observed: (a) decoupled mode and (b) coupled mode.

Fig. 3.Network representation of the plenum, burners, and chamber (PBC configuration).

dynamics is described using standard linearized acoustics for per-fect gases in the low-Mach-number-approximation. The mean flow induced by swirlers remains slow[31], and azimuthal waves prop-agate at the sound speed, which is different in the plenum (fresh gas at sound speed c0

u) and the chamber (burnt gas at sound speed

c0_).

In the initial ATACAMAC study of Parmentier et al.[22], a BC (burners + chamber) configuration was considered: an annular chamber is fed by N burners without taking into account the exis-tence of a plenum; the impedance imposed at the inlet of the N burners corresponded to a closed end ðw0_{¼ 0Þ, and flames were}

lo-cated at the burners, extremity (zf ;i’ Li, where Liis the length of

the ith burner). This section describes how this BC case is extended to PBC (plenum + burners + chamber) configurations where an annular plenum feeds N ducts (‘‘burners’’) that are all connected to an annular chamber. Most annular gas turbine chambers can be modeled using this scheme (Fig. 3).

In a PBC configuration, burners are connected at both ends to the annular plenum and chamber. The boundary conditions w0_{¼ 0 used for BC configurations are retained here only on walls}

of the annular cavities, not at the inlet/outlet sections of the burn-ers. Mean density in the annular chamber is denoted

q

0₍

_q

0

uin the

plenum). The subscript u stands for unburnt gases. The perimeter and the section of the annular plenum are denoted 2Lp¼ 2

p

Rp

and Sp, respectively, while Li and Si stand for the length and

cross-section area of the ith burner. The perimeter and the section of the annular chamber are denoted 2Lc¼ 2

p

Rc and Sc,

respec-tively. The position along the annular plenum and chamber is given by the angle h defining abscissa xp¼ Rph for the plenum and

xc¼ Rchfor the chamber. The location of the flames in the burners

is given by the normalized abscissa

a

¼ zf ;i=Li(Fig. 3).

2.2. Acoustic waves description and ANR methodology

To reduce the size of the system, a new methodology called ANR (annular network reduction) is proposed to extract only useful information on azimuthal modes of the resonant combustor. First, the combustor is decomposed into N sectors (Fig. 4) by assuming that every sector can be studied separately and that no flame-to-flame interaction occurs between neighboring sectors, a question that is still open today[32,33]. Staffelbach et al.[31]have shown that this was the case in LES of azimuthal modes. Worth and

Daw-son [4,34] have also demonstrated experimentally that this

assumption in annular combustors is valid when the distance be-tween burners is large enough to avoid flames merging.

For each individual sector, the acoustic problem may be split into two parts: propagation (Section2.2.1 and already described

in [22,35,36]) and H-shaped connector (Section 2.2.2) (Fig. 5).

The angle h and the coordinates of the plenum xpand chamber xc

take their origin at the burner i. The end of the sector i is located at xp¼2L_Np and xc¼2L_Nc(thus assuming that burners are evenly

lo-cated around the annular combustor). H-shaped connectors are as-sumed to be compact regarding the acoustic wavelength. 2.2.1. Propagation

There are two main paths to analyzing azimuthal modes in annular chambers with simple models. The first is to write the mode explicitly as a function of the azimuthal angle h, as done in

[7,8], for example. The second approach is to use a network

discret-ization along the azimuthal direction[1,22]. An advantage of the first method is the capability of considering modes that are azi-muthal but also involve an axial dependance, something that can-not be done yet with the second approach. However, the second method describes explicitly the coupling between annular cavities and burners and therefore has been applied here for ATACAMAC.

Assuming linear acoustics, the pressure and velocity perturba-tions inside the ith sector of the annular chamber (denoted via sub-script c; i) can be written as

p0

c;iðxc; tÞ ¼ ðAicosðkxcÞ þ BisinðkxcÞÞe

ÿjxt_{; ð1Þ}

q

0_c0_u0

c;iðxc; tÞ ¼ jðAisinðkxcÞ ÿ BicosðkxcÞÞeÿjx t

; ð2Þ

where j2¼ ÿ1; k ¼

x

=c0_{is the wavenumber, and A}

iand Biare

com-plex constants.

From Eqs.(1) and (2), pressure and velocity perturbations at two positions in the chamber xc0and xc0þDxcare linked by

p0 c;i 1 j

q

0c0u 0 c;i " # ðxc0þDxc;tÞ ¼ cosðk

D

xcÞ ÿ sinðk

D

xcÞ sinðk

D

xcÞ cosðk

D

xcÞ   |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} RðkDxcÞ p0 c;i 1 j

q

0c0u 0 c;i " # ðxc0;tÞ : ð3Þ

The matrix RðkDxcÞ is a 2D rotation matrix of angle kDxc. In the

case where N burners are equally distributed over the annular chamber, the propagation in the chamber between burners is fully described by the transfer matrix R k2Lc

N

ÿ
_{, where 2L}

cis the chamber

perimeter.

Wave propagation in the ith sector of the plenum satisfies the same equation as in the chamber if the sound speed in fresh gases c0 uis used, p0 p;i 1 j

q

0uc0uu0p;i " # ðx0þDxp;tÞ ¼ cosðku

D

xpÞ ÿ sinðku

D

xpÞ sinðku

D

xpÞ cosðku

D

xpÞ   |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} RðkuDxpÞ p0 p;i 1 j

q

0uc0uu0p;i " # ðx0;tÞ ; ð4Þ where ku¼

x

=c0u.

Fig. 4.Decomposition of an annular combustor into N sectors.

Fig. 5.ANR methodology: splitting of one sector into propagation and H-shaped section parts for a PBC configuration.

Propagation in the ith burner from the plenum (z ¼ 0) to the flame (z ¼

a

Li) and from z ¼

a

Lito z ¼ Li(Fig. 6) can be described

in the same way, assuming propagation in fresh gases for the first part (0 < z <

a

Li) and in hot gases in the second part (

a

Li< z < Li):

Therefore, equations for wave propagation in the ith burner are

p0 i 1 j

q

0uc0uw0i " # ðaLi;tÞ ¼ Rðku

a

LiÞ p 0 i 1 j

q

0uc0uw0i " # ð0;tÞ ð5Þ and p0 i 1 j

q

0c0w 0 i " # ðLi;tÞ ¼ Rðkð1 ÿ

a

ÞLiÞ p 0 i 1 j

q

0c0w 0 i " # ðaLi;tÞ : ð6Þ 2.2.2. H-shaped connector

The physical parameters at the entrance (located at the end of the ði ÿ 1Þth sector corresponding to h ¼2p

N and consequently

xp¼2LNpand xc¼2LNc) must be linked to the output parameters

(cor-responding to the beginning of the ith sector located at h ¼ 0 and consequently xp¼ xc¼ 0), as shown inFig. 7.

The H-shaped connectors are treated as compact elements in the azimuthal direction. Jump conditions are first written along the x direction at z ¼ 0 and z ¼ Li: at low Mach number[2], they

correspond to the continuity of pressure and volume rate through the interface. For the sake of simplicity, more complex interactions (e.g., effective length or pressure drop effects)[20]are neglected, knowing they could affect the coupling mechanisms between cav-ities. At z ¼ Li, p0 c;iÿ1 xc¼ 2Lc N ; t   ¼ p0

c;iðxc¼ 0; tÞ ¼ p0iðz ¼ LiÞ; ð7Þ

Scu0

c;iÿ1 xc¼ 2Lc

N ; t  

þ Siw0_{iðz ¼ LiÞ ¼ Scu}0

c;iðxc¼ 0; tÞ; ð8Þ and at z ¼ 0, p0 p;iÿ1 xp¼ 2Lp N ; t   ¼ p0 p;iðxp¼ 0; tÞ ¼ p 0 iðz ¼ 0Þ; ð9Þ Spu0 p;iÿ1 xp¼ 2Lp N ; t   ¼ Siw0 iðz ¼ 0Þ þ Spu 0 p;iðxp¼ 0; tÞ: ð10Þ

Jump conditions are also required in the burners through the flames located at z ¼ zf ;i. They are assumed to be planar and

compact: their thickness is negligible compared to the acoustic wavelength. Flames are located at z ¼ zþ

f ;i¼ zÿf ;i’

a

Li, where

super-scripts + and ÿ denote the downstream and upstream positions of the ith flame. At low Mach number, jump conditions through the flame imply equality of pressure and flow rate discontinuity due to an extra volume source term related to unsteady combustion

[2], p0 i zþf ;i   ¼ p0 i zÿf ;i   ; ð11Þ Siw0 i zþf ;i   ¼ Siw0 i zÿf ;i   þ

c

uÿ 1

c

uP 0

X

_ 0 T;i; ð12Þ

where P0is the mean pressure and

c

uis the heat capacity ratio of

fresh gases. The unsteady heat release _X0

T;iis expressed using the

FTF model (flame transfer function)[37],

c

uÿ 1

c

uP0 _

X

0 T;i¼ Siniejxsiw0i zÿf ;i   ; ð13Þ

where the interaction index ni1and the time delay

s

iare input data

(depending on frequency) describing the interaction of the ith flame with acoustics. Therefore, the jump condition in Eq.(12)is recast using Eq.(13): Siw0 i zþf ;i   ¼ Sið1 þ niejxsi_Þw0 i zÿf ;i   : ð14Þ

Thanks to Eqs.(7)–(14), a transfer matrix Ti of the H-shaped

connector ofFig. 7is defined as

p0_{pðxp¼ 0; tÞ} 1 j

q

0uc0uu0pðxp¼ 0; tÞ p0 cðxc¼ 0; tÞ 1 j

q

0c0u0cðxc¼ 0; tÞ 2 6 6 6 6 4 3 7 7 7 7 5 i ¼ Ti p0_pðxp¼2Lp N ; tÞ 1 j

q

0uc0uu0p xp¼ 2Lp N; t   p0 c xc¼2LNc; t ÿ
1 j

q

0c0u0cðxc¼2LNc; tÞ 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 iÿ1 ; ð15Þ

where the transfer matrix Tiis

Ti¼ Idþ 2 0 0 0 0

C

i;1 0

C

i;2 0 0 0 0 0

C

i;3 0

C

i;4 0 2 6 6 6 4 3 7 7 7 5 ð16Þ

Fig. 6.Propagation in the ith burner for a PBC configuration in the ANR method.

Fig. 7.H-shaped overview in the ANR method. 1_{Typical values of the interaction index n}

and the coefficientsCi;k; k¼ 1—4, are

C

i;1¼ ÿ 1 2 Si Sp

cosðkð1 ÿ

a

ÞLiÞ cosðku

a

LiÞ ÿ F sinðkð1 ÿ

a

ÞLiÞ sinðku

a

LiÞ cosðkð1 ÿ

a

ÞLiÞ sinðku

a

LiÞ þ F sinðkð1 ÿ

a

ÞLiÞ cosðku

a

LiÞ;

ð17Þ

C

i;2¼ 1 2 Si Sp 1

cosðkð1 ÿ

a

ÞLiÞ sinðku

a

LiÞ þ F sinðkð1 ÿ

a

ÞLiÞ cosðku

a

LiÞ; ð18Þ

C

i;3¼ 1 2 Si Sc F

cosðkð1 ÿ

a

ÞLiÞ sinðku

a

LiÞ þ F sinðkð1 ÿ

a

ÞLiÞ cosðku

a

LiÞ; ð19Þ

C

i;4¼ ÿ 1 2 Si Sc

F cosðkð1 ÿ

a

ÞLiÞ cosðku

a

LiÞ ÿ sinðkð1 ÿ

a

ÞLiÞ sinðku

a

LiÞ cosðkð1 ÿ

a

ÞLiÞ sinðku

a

LiÞ þ F sinðkð1 ÿ

a

ÞLiÞ cosðku

a

LiÞ;

ð20Þ

with the flame parameter F:

F ¼

q

0_c0

q

0 uc0u

ð1 þ nejxs_{Þ: ð21Þ}

These coefficients are the coupling parameters for PBC configu-rations. Ci;1 and Ci;2 are linked to the plenum/burner junction

(depending on Si=Sp, which measures the ratio between the burner

section Siand the plenum section Sp), whileCi;3andCi;4are linked

to the chamber/burner junction (depending on Si=Sc, which

mea-sures the ratio of the burner section to the chamber section (Fig. 3)).

2.3. Dispersion relation calculation given by the ANR method In previous sections, the overall problem has been split into smaller parts, and now it has to be reconstructed to obtain the dis-persion relation for the whole system. First, the pressure and veloc-ity fluctuations at the end of the ði ÿ 1Þth sector are linked to those at the end of the ith sector using Eqs.(3), (4), and (15),

p0 p xp¼ 2Lp N; t   1 j

q

0uc0uu0p xp¼ 2Lp N; t   p0 c xc¼2LNc; t ÿ
1 j

q

0c0u0c xc¼2LNc; t ÿ
2 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 5 i ¼ Ri p0 pðxp¼ 0; tÞ 1 j

q

0uc0uu0pðxp¼ 0; tÞ p0_{cðxc¼ 0; tÞ} 1 j

q

0c0u0cðxc¼ 0; tÞ 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 i ¼ RiTi p0 p xp¼ 2Lp N ; t   1 j

q

0uc0uu0pðxp¼ 2Lp N ; tÞ p0 c xc¼2LNc; t ÿ
1 j

q

0c0u 0 c xc¼2LNc; t ÿ
2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 iÿ1 ; ð22Þ

where Tiis the matrix defined in Eq.(16)and Riis the propagation

matrix inside the plenum and chamber in the ith sector, defined by

Ri¼ R ku2Lp N   0 0 0 0 0 0 0 0 R k 2Lc N ÿ
2 6 6 6 4 3 7 7 7 5; ð23Þ where R ku2L_Np   and R k2Lc N

ÿ
_{are 2-by-2 matrices defined in Eqs.}

(3) and (4)

Then Eq.(22)can be repeated through the N sectors, and peri-odicity imposes that

p0 p xp¼ 2Lp N ; t   1 j

q

0uc0uu0p xp¼ 2Lp N ; t   p0 c xc¼2LNc; t ÿ
1 j

q

0c0u0c xc¼2LNc; t ÿ
2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 i¼1 ¼ Y 1 i¼N RiTi ! p0 p xp¼ 2Lp N; t   1 j

q

0uc0uu0p xp¼ 2Lp N; t   p0 c xc¼2LNc; t ÿ
1 j

q

0c0u0c xc¼2LNc; t ÿ
2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 i¼1 : ð24Þ

System Eq.(24) leads to non-null solutions if and only if its determinant is null, det Y 1 i¼N RiTiÿ Id ! ¼ 0; ð25Þ

where Idis the 4-by-4 identity matrix and the matrix M ¼Q1i¼NRiTi

is the transfer matrix of the overall system. Eq.(25)provides an im-plicit equation for the pulsation

x

and gives the stability limits and the frequency of unstable modes. Note that Eq.(25)given by the ANR methodology only involves a 4-by-4 determinant2_irrespective

of the number of burners N.

3. Analytical procedure and coupling limits

Due to significant nonlinearities, Eq.(25)cannot be solved ana-lytically in the general case. However, different situations (Fig. 8) can be exhibited where Eq.(25)can be solved, depending on values taken by the coupling parameters }p;iand }c;i:

}p;i¼ maxðj

C

i;1j; j

C

i;2jÞ ð26Þ

and

}c;i¼ maxðj

C

i;3j; j

C

i;4jÞ: ð27Þ

The parameters }p;iand }c;imeasure the coupling effect of the

plenum/burner junction and the chamber/burner junction, respec-tively, for the ith sector. They depend only on the geometry (sec-tion ratios Si=Sp and Si=Sc as well as the burner’s length Li) and

the flame (the flame interaction factor F and the flame position

a

). Longitudinal modes in the burners can be obtained with this model, but only purely azimuthal modes will be studied in this pa-per (Fig. 8).

3.1. Fully decoupled situations (FDPp and FDCp)

If the coupling parameters have vanishing small values (}p;i¼ 0

and }c;i¼ 0), the coupling matrices of each sector (Ti in Eq.(25))

are the identity matrix. As a consequence, the dispersion relation (Eq.(25)) reduces to det Y 1 i¼N Riÿ Id ! ¼ 0: ð28Þ

The matrices Ribeing block matrices of 2-by-2 rotation matrices

that satisfy Rðh1ÞRðh2Þ ¼ Rðh1þ h2Þ, Eq.(28)becomes

det Y 1 i¼N Riÿ Id ! ¼ det Rð2kuLpÞ ÿ Id 0 0 0 0 0 0 0 0 Rð2kLcÿ IdÞ 2 6 6 6 4 3 7 7 7 5 0 B B B @ 1 C C C A

¼ detðRð2kuLpÞ ÿ IdÞ detðRð2kLcÞ ÿ IdÞ ¼ 0: ð29Þ

2 _{The ANR methodology retains only useful information related to azimuthal}

modes of the annular cavities. Knowing that these modes are a combination of two characteristic waves, the minimum size of the matrix system is 2C-by-2C, where C is the number of annular cavities (here C ¼ 2: plenum + chamber).

Solutions of Eq.(29)are kuLp¼ p

p

and kLc¼ p

p

, where p is an

integer. They correspond to two families of modes that live sepa-rately (case A inFig. 8):

 FDPp—fully decoupled plenum mode of order p: They satisfy the relation kuLp¼ p

p

and correspond to the azimuthal modes

of the annular plenum alone (Fig. 8).

 FDCp—fully decoupled chamber mode of order p: They satisfy the relation kLc¼ p

p

and correspond to the azimuthal modes of

the annular chamber alone (Fig. 8).

3.2. Chamber or plenum decoupled situations (PBp and BCp) These situations correspond to cases where either the plenum or the chamber is fully decoupled from the burners, i.e., respec-tively }p¼ 0 or }c¼ 0, (cases B and C inFig. 8). The transfer

matri-ces of each sector are block-triangular, leading to a simple relation for the determinant and consequently for the dispersion relation (Eq.(25)) without assumption on the non-null parameter }p or

}c. As for the fully decoupled situations, two families of modes

can be exhibited (cases B and C inFig. 8):

 PBp—plenum/burners mode of order p: In this situation, }c¼ 0. The annular chamber is fully decoupled from the system

(burners + plenum) and the dispersion relation Eq.(25)reduces to two equations: detðRð2kLcÞ ÿ 1Þ ¼ 0 ð30Þ det Y 1 i¼N R 2kuLp N   _{1 0}

C

i;1 1   ÿ Id ! ¼ 0: ð31Þ

Eq.(30) corresponds to the dispersion relation of a FDCp mode, while Eq.(31)is the dispersion relation of a PB (plenum + burners) configuration where an impedance Z ¼ 0 is imposed at the down-stream end of the burner, simulating the large chamber decoupled from the system (case B inFig. 8). This latter mode is referred to as ‘‘PBp’’ standing for plenum/burners mode of order p. These situa-tions are, however, unrealistic because they neglect all interacsitua-tions between the annular plenum and chamber: the only solution to obtaining }c! 0 in a PBC configuration is an infinite cross section

of the chamber (Sc! 1 in Eqs.(19) and (20)). Therefore, this paper

will focus on other situations (e.g., cases D and E inFig. 8) that are more representative of real engines by including the interaction be-tween annular cavities.

 BCp—burners/chamber mode of order p: In this situation, }p¼ 0, so the annular plenum is fully decoupled from the rest

of the system and the dispersion relation Eq.(25)reduces to two equations: detðRð2kuLpÞ ÿ 1Þ ¼ 0 ð32Þ det Y 1 i¼N R 2kLc N   _{1 0}

C

i;4 1   ÿ Id ! ¼ 0: ð33Þ

The first equation (Eq. (32)) is the dispersion relation of a FDPp mode, while Eq. (33) is the dispersion relation of a BC (burners + chamber) configuration where a pressure node (Z ¼ 0) is imposed at the upstream end of the burner, modeling the large plenum decoupled from the burners and the annular chamber (case C inFig. 8). This latter mode is called ‘‘BCp’’ for burners/cham-ber mode of order p and was already studied by Parmentier et al.

[22].

3.3. Weakly coupled situations (WCPp and WCCp)

When }pand }care both nonzero, both azimuthal modes of the

plenum and the chamber can exist and interact through the burn-ers. If }pand }cremain small (Eq.(34)), an asymptotic solution can

be constructed:

8

i; 0 < }_p;i 1 and 0 < }c;i 1: ð34Þ

The parameters }p;i (Eq.(26)) and }c;i (Eq.(27)) measure the

strength of the coupling effect of the plenum/burner junction and the chamber/burner junction, respectively, for the ith sector. If they are small, the transfer matrices of each sector (TiRiin Eq.(25)) are

close to the rotation matrix Ridefined in Eq.(4), so that the

eigen-frequencies of the system will be close to a FDPp or a FDCp mode. Consequently, as for fully decoupled modes, two families of modes can be exhibited (case D inFig. 8):

 WCPp—weakly coupled plenum mode of order p: This mode is close to a FDPp mode, and the solution for the wavenumber kucan be searched for as an expansion around this case,

kuLp¼ p

p

þ



p; ð35Þ

where



p p

p

.

 WCCp—weakly coupled chamber mode of order p: This mode is close to a FDCp mode, and the solution for the wavenumber k can be searched for as an expansion around this case,

kLc¼ p

p

þ



c; ð36Þ

where



c p

p

.

For weakly coupled modes, this low-coupling assumption al-lows a Taylor expansion of the dispersion relation (Eq.(25)), which can be truncated and solved, providing analytical solutions for



c

and



p. This expansion is case-dependent: The N ¼ 4 case will be

detailed in Section5. Basically, results show that azimuthal modes will be either chamber or plenum modes slightly modified by their interaction with the rest of the combustor.

Note that the low-coupling assumption (}p;i 1 and }c;i 1)

does not imply low thermoacoustic coupling (ni 1) because

sur-face ratios between burner and plenum or chamber are usually small (Si=Sc 1 and Si=Sp 1).

In the specific configuration where the flames are located at the end of the burner (

a

¼ zf ;i=Li¼ 1 inFig. 3), however, the coupling

parameters simplify as

C

i;1¼ ÿ 1 2 Si Sp cotanðkuLiÞ; ð37Þ

C

i;2¼ 1 2 Si Sp 1 sinðkuLiÞ; ð38Þ

C

i;3¼ 1 2 Si Sc F sinðkuLiÞ; ð39Þ

C

i;4¼ ÿ 1 2 Si ScF cotanðkuLiÞ; ð40Þ

where F is the flame parameter defined in Eq.(21). Eqs.(37)–(40)

correspond to an extension of the coupling parameters proposed by Schuller et al.[26]for longitudinal instabilities and Parmentier et al.[22] for azimuthal instabilities in a BC configuration. They show that decoupling (}p;i 1 and }c;i 1) can be expected in this

case for small section ratios Si=Sp 1 and Si=Sc 1, when the flame

parameter F (Eq.(21)) is small too. 3.4. Strongly coupled situations (SCp)

The low-coupling assumption (}p;i 1 and }c;i 1) is not

va-lid at high flame-interaction factors F ¼q0_c0 q0

uc0uð1 þ ne

jxs_Þ

 

or high surface ratios (Si=Sp or Si=Sc). In these situations (case E inFig. 8),

a numerical resolution of the analytical dispersion relation (Eq.

(25)) is required. It can be achieved by a nonlinear solver based on the Newton–Raphson algorithm.

No rule already exists to distinguish a weakly or strongly cou-pled situation for real engines (characterized by an unknown crit-ical parameter }crit,Fig. 8). Moreover, classifying modes into two

families, as is the case for fully decoupled situations (FDPp and FDCp modes), plenum or chamber decoupled situations (PBp and BCp modes), and weakly decoupled situations (WCPp and WCCp modes), is not possible any more due to the interaction between all parts of the system. A first attempt to identify key parameters and rules to differentiate weakly and strongly coupled situations is described in Section6.

In the remaining of the paper, the weakly and strongly coupled situations (Fig. 8) will be studied in the PBC configuration with four burners (N ¼ 4) described in Section4. The transition from weakly

(Section5) to strongly coupled (Section6) regimes is controlled by a critical coupling limit factor }crit. The transition occurs when

maxð}p; }cÞ > }crit. The geometry being fixed (Table 1) and the

cou-pling parameters (}pand }c) depending only on the geometry and

the flame, the transition will be triggered by increasing the flame interaction index ni of the flames from ni¼ 1:57 (weak coupling)

to ni¼ 8:0 (strong coupling).

4. Validation in a simplified model chamber

Eigenfrequencies and mode structures of the analytical resolu-tion of the dispersion relaresolu-tion under the low-coupling-factors assumption (Fig. 8, all modes except SCp) are first compared to a full 3D acoustic code and to the direct resolution of Eq. (25)in the case of a simplified 3D PBC configuration, which is used as a toy model for ATACAMAC and corresponds to a typical industrial gas turbine.

4.1. Description of the simplified PBC configuration

The 3D geometry (Fig. 9) corresponds to a PBC setup with N ¼ 4 burners, similar toFig. 3(characteristics defined inTable 1). The mean radii Rpand Rcof the cylindrical chamber and plenum are

de-rived from the half perimeter Lp and Lc of the analytical model.

Boundary conditions correspond to impermeable walls everywhere.

4.2. Description of the 3D acoustic code

To validate the assumptions used in the ATACAMAC formula-tion, it is interesting to compare its results with the output of a full 3D acoustic solver. Here, AVSP was used. AVSP is a parallel 3D code devoted to the resolution of acoustic modes of industrial combus-tion chambers[38]. It solves the eigenvalues problem issued from a discretization on unstructured meshes of the Helmholtz equation with a source term due to the flames. The mesh used here (Fig. 9, left) is composed of 230,000 cells, which ensures grid indepen-dence. The flame–acoustic interaction is taken into account via the FTF model[37] similarly to the expression used in Eq.(13). The local reaction term is expressed in burner i as

_

x

i¼ nu;iejxsiw0ðxref;iÞ: ð41Þ

The local interaction index nu;idescribes the local

flame–acous-tic interactions. The values of nu;iare assumed to be constant in the

flame zone i (Fig. 9) and are chosen to recover the global value of interaction index ni of the infinitely thin flame when integrated

over the flame zone i[38]. They are also assumed to be indepen-dent of frequency for simplicity. Heat release fluctuations in each flame zone are driven by the velocity fluctuations at the reference points xref;i located in the corresponding burner. In the infinitely

thin flame model these reference points are the same as the flame locations zf. In AVSP, the reference points were placed a few

milli-meters upstream of the flames (Fig. 9) to avoid numerical issues. This was proved to have only a marginal effect on the computed frequency[38,39].

4.3. Construction of a quasi-1D network from a real 3D combustor A quasi-one-dimensional model of a simplified PBC configura-tion described in Secconfigura-tion4.1can be constructed usingTable 1. Even though the present model is quasi-one-dimensional, simple correc-tions can be incorporated to capture 3D effects.

First, the burners considered inFig. 9are long narrow tubes for which end effects modify acoustic modes. In the low-frequency range, this can be accounted for [40] and a standard length Table 1

Parameters used for numerical applications. Chamber Half perimeter Lc 6.59 m Section Sc 0.6 m2 Plenum Half perimeter Lp 6.59 m Section Sp 0.6 m2 Burner Length L0 i 0.6 m Section Si 0.03 m2 Fresh gases Mean pressure p0 _{2  10}6 _Pa Mean temperature T0 u 700 K Mean density q0 u 9.79 kg=m3

Mean sound speed c0

u 743 m=s Burnt gases Mean pressure p0 2  106 Pa Mean temperature _T0 _{1800 K} Mean density q0 3.81 kg=m3

Mean sound speed c0 _{1191 m=s}

Flame parameters

Interaction index ni Variable –

Time delay si Variable s

Thickness efl 0.03 m

correction for a flanged tube[41]is applied at the two burner’s ends. The corrected length Lifor the burners is

Li¼ L0i þ 2  0:4 ffiffiffiffiffiffiffiffiffiffiffiffi 4Si=

p

p ; ð42Þ where L0

i is the ith burner length without correction and Siis the

surface of the ith burner.

Second, the position of the compact flames is defined via the parameter

a

¼ zf ;i=Li, where Li is the corrected burner length and

zf ;i is the position of the center of the flame. In Sections5 and 6,

the flame position parameter is set to

a

¼ 0:88, a location chosen because it is far away from all pressure nodes.

5. Mode analysis of a weakly coupled PBC configuration with four burners (N ¼ 4)

Under the weak-coupling-factors assumption (0 < }p;i 1 and

0 < }c;i 1), frequencies of the whole system can be analyzed

assuming small perturbations around the chamber alone (FDCp mode at k0Lc¼ p

p

) or plenum alone (FDPp mode at k0uLp¼ p

p

),

wavenumbers leading to two families of modes that appear sepa-rately (Section3.3),

kLc¼ p

p

þ



cðWCCp modesÞ ð43Þ or

kuLp¼ p

p

þ



pðWCPp modesÞ; ð44Þ

where p is the mode order,



c p

p

, and



p p

p

. Since the two

families of modes behave in the same manner (only radius, density, and sound speed are changed), only weakly coupled chamber modes will be detailed in this section.

Due to symmetry considerations of the case N ¼ 4, odd-order modes ðp ¼ 2q þ 1; q 2 NÞ, and even-order modes ðp ¼ 2q; q 2 NÞ will not behave in the same manner and are analyzed in Sections

5.1 and 5.2, respectively.

5.1. Odd-order weakly coupled modes of the PBC configuration with four burners (N ¼ 4)

Considering odd-order modes ðp ¼ 2q þ 1; q 2 NÞ with the low-coupling-limit assumption (0 < }p;i 1 and 0 < }c;i 1) leads to

the expansion kLc¼ p

p

þ



c(for a WCCp mode). A Taylor

expan-sion can therefore be used to obtained a simplified analytical expression for the transfer matrix of the ith sector and conse-quently for the dispersion relation (Eq.(25)). This approach is fully detailed inAppendix Afor the first mode on a single-burner case (N ¼ 1) for simplicity. The same approach for the pth mode and four burners (N ¼ 4) leads (seeTable A.1) to

sin ðp

p

bÞ2



2 cþ 4



c

C

0₄þ 4

C

0₄ 2 h i þ o



2 c ÿ
¼ 0; ð45Þ where b ¼c0Lp c0 uLcandC 0

4is the value ofC4when kLc¼ p

p

. Note that all

the burners share the same length and cross section in this config-uration, so the index i of Eq.(20)was removed for simplicity.

The b parameter can be viewed as a tuning parameter between cavities: it compares the period of the azimuthal modes in the ple-num alone

s

0 p¼ 2Lp pc0 u  

and in the chamber alone

s

0 c¼2Lpc0c

 

. In gen-eral, the two annular volumes are not tuned and the periods

s

0

pand

s

0

cof the azimuthal modes of the plenum and the chamber do not

match; i.e., b ¼

s

0

p=

s

0c–l; l 2 N (for example, for the first chamber

and plenum modes (p ¼ 1) ofTable 1where b ’ 1:60), so that the only solution to satisfy Eq.(45)is



2 cþ 4



c

C

0₄þ 4

C

0₄ 2 þ o



2 c ÿ
¼ 0: ð46Þ

This quadratic equation has a double root,3

Fig. 9.3D toy model to validate the ATACAMAC methodology. Perfect annular chamber and plenum connected by four burners (N ¼ 4). : Line in the plenum (PL) and in the chamber (CL) along which absolute pressure and phase will be plotted.

3_{This approach can be extended at higher orders to unveil plenum/chamber}

interactions. From a third-order truncation of the dispersion relation (Eq.(25)), a second-order correction on eigenfrequencies found in Eq.(47)is obtained for WCCp modes,



c¼ ÿ2

C

0₄ÿ HðbÞ

C

0₂

C

0₃ðWCCp modesÞ;



c¼ ÿ2

C

0₄; ð47Þ

whereC04is the value ofC4(Eq.(20)) at

x

¼

x

0¼ p

p

c0=Lc.

Real and imaginary parts of the frequency obtained in Eq.(47)

are compared with the exact numerical resolution of the dispersion relation Eq. (25) and with AVSP results in Fig. 10. Very good agreement, is found showing that the asymptotic expression for the frequency (Eq.(47)) is correct.

From Eq.(47), a simple analytical stability criterion can be de-rived as explained in Appendix B for weakly coupled chamber modes (Eq.(B.3)), sin 2

p

s

s

0 c   |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} effect ofs sin 2p

p

a

Lic0 Lcc0 u   |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} effect ofa < 0 ðWCCp modesÞ; ð48Þ where

s

0 c¼2c 0 pLc.

Time delay

s

of the FTF and flame position

a

have an effect on the stability and can be studied separately:

 Time delay—From Eq.(48), a critical time delay controlling the transition from stable to unstable regimes can be obtained:

s

crit¼

s

0c=2¼ Lc

pc0for WCCp modes. This result being also valid

for plenum modes (i.e., WCPp modes; see Eq.(B.4)leading to

the critical time delay

s

0

p=2¼

Lp

pc0

u), stability ranges of the two

first azimuthal modes WCC1 and WCP1 can be displayed simul-taneously (Fig. 11). This shows that the region where both the chamber and plenum modes are stable is smaller than the sta-bility ranges of modes taken separately: stabilizing one mode of the system cannot ensure the stability of the whole system. Of course, this result does not include any dissipation or acoustic fluxes through the boundaries[42], which would increase sta-bility regions.

 Flame position—Similarly to longitudinal modes in the Rijke

tube[2,26,43], the flame position (defined by

a

) also controls

the stability (Eq.(48)).Appendix Cdemonstrates an analytical expression for the critical flame position for weakly coupled modes (Eq. (C.2)) at which the stability change occurs. These expressions are close to the solution found in Rijke tubes (Eq.

(C.1)) and have been validated for several weakly coupled

modes (Table C.1andFig. C.1).

5.2. Even-order modes of the PBC configuration with four burners (N ¼ 4)

Assuming even-order weakly coupled chamber modes ðp ¼ 2q; q 2 NÞ and using Eq. (25) with the low-coupling-limit assumption ð0 < }p;i 1 and 0 < }c;i 1Þ, the dispersion relation

(Eq.(25)) becomes (Table A.1)

sin ðp

p

b=2Þ2



2 cþ 4



c

C

04 h i þ o



2 c ÿ
¼ 0; ð49Þ where b ¼kuLp kLc andC 0

4is the value ofC4when kLc¼ p

p

.

When chamber and plenum frequencies do not match, i.e., b – 1 (for example, for the second (p ¼ 2) mode of Table 1, where b’ 1:60), the only solution to satisfying Eq.(49)is to have



2

cþ 4



c

C

04þ o



2c ÿ

¼ 0: ð50Þ

This quadratic equation has two distinct roots:



c¼ 0 and



c¼ ÿ4

C

0₄: ð51Þ

The first solution (



c¼ 0) of Eq.(51)corresponds to modes that

are not affected by the flames: the symmetry of the four-burners case (N ¼ 4) allows even-order modes to impose a pressure node at each burner location, suppressing the flame effect on these modes, which become neutral.

The second solution of Eq. (51)



c¼ ÿ4C04

 

correspond to modes that are strongly affected by the flame because it imposes a pressure anti-node (maximum pressure) downstream of each burner.

Real and imaginary parts of the frequency obtained from Eq.

(51)are compared with the numerical solutions of the dispersion relation Eq. (25) and to AVSP results in Fig. 12. Very good Fig. 10.Eigenfrequency of the WCC1 mode for four burners (N ¼ 4) with ni¼ 1:57 as a function ofs=s0c. : Numerical resolution of the dispersion relation Eq.(25). :

Analytical model prediction Eq.(47). : AVSP results.

Fig. 11.Stability maps (Eq.(48)) of the two first azimuthal modes (p ¼ 1) for a flame located ata’ 0:88: WCP1 (top), WCC1 (middle), and both WCP1 and WCC1 (bottom).

agreement is found and the nonperturbed mode (



c¼ 0) is

cor-rectly captured (straight lines).

Using Eq.(51)andAppendix B leads to an analytical stability criterion for WCCp even-order modes (Eq.(B.3)). This result being also valid for WCPp modes (Eq.(B.4)) and using the results from Section 5.1, stability maps of the weakly coupled plenum and chamber modes (WCC1, WCC2, WCP1 and WCP2) can be plotted together (Fig. 13), highlighting the difficulty of getting a stable sys-tem in the absence of dissipation, as assumed here: considering only these four modes, no stable region is found for time delays lower than 19.1 ms in the case described inTable 1(Fig. 13).

5.3. Mode structure of weakly coupled modes

In weakly coupled situations, acoustic activity is present only in one annular cavity and in the burners, as shown inFig. 14for the WCC1 mode (the same mode with the opposite rotation direction is also captured by AVSP but not shown here).Figure 15shows that this mode is purely rotating in the chamber, while it is mixed in the plenum. Therefore, the combination of the two weakly coupled chamber modes with the clockwise rotation (shown inFig. 15) and the counter clockwise rotation (not shown) make it possible to have purely rotating or purely standing modes, but not neces-sarily in the two annular cavities at the same time.

5.4. Stability map of weakly coupled situations

Sections 5.1 and 5.2 focused on weakly coupled situations where the low-coupling-factor assumption (}p 1 and }c 1)

is valid. Stability maps of perturbed modes (i.e.,



–_{0) in the} com-plex plane ½Reðf Þ; Imðf ފ are well suited to show differences be-tween the several regimes—weakly coupled and strongly coupled situations. These maps are oriented circles ( ) (WCP1) and (WCC1) inFig. 16) centered at the frequency f ðni¼ 0Þ ’ f0,

which corresponds to a passive flame mode (ni¼ 0 :  inFig. 16)

and is approximately the frequency of the cavity alone f0¼pc

0

2L

 

. The radius is proportional to the coupling factor, which can be in-creased via the surface ratios (Si=Spand Si=Sc) or the flame

interac-tion index ni. In the weakly coupled regime, WCCp and WCPp

modes do not strongly interact, as shown in Fig. 16: modes in the plenum and in the chamber live separately.

Fig. 12.Eigenfrequency of the WCC2 mode for four burners (N ¼ 4) with ni¼ 1:57 as a function ofs=s0c. : Numerical resolution of the dispersion relation Eq.(25). :

Analytical model prediction Eq.(51). : AVSP results.

Fig. 13.Stability maps of the WCC1, WCC2, WCP1, and WCP2 modes.

Fig. 14.Pressure mode structures (p0_{¼ jp}0_{j cosðargðp}0_{ÞÞ) obtained with AVSP with}

pressure isolines (left) and pressure along the azimuthal direction in the annular chamber ( ) and annular plenum ( ) for the WCC1 mode of a PBC configuration with four burners (N ¼ 4) and ni¼ 1:57 at the time delays=s0c¼ 0.

Fig. 15.Pressure phase of the WCC1 mode obtained with AVSP at ni¼ 1:57 along

the azimuthal direction in the annular chamber ( ) and the annular plenum ( ) for time delays=s0

6. Mode analysis of a strongly coupled PBC configuration with four burners (N ¼ 4)

For weakly coupled cases (Section5), the frequencies of the azi-muthal modes in the plenum and in the chamber are only margin-ally affected by the flame (Figs. 10 and 12), so they can never match (Fig. 16). However, if the flame interaction index (ni) is

lar-ger, the frequencies of the azimuthal modes of the annular plenum and the annular chamber change more and the possibility of hav-ing the two frequencies match opens an interesthav-ing situation where the whole system can resonate. This corresponds to the strongly coupled modes of order p (referred as ‘‘SCp’’ modes) stud-ied in this section.

Figure 17 shows the stability maps of the WCC1 ( ) and

WCP1 ( ) modes obtained with low flame-interaction index (ni¼ 5:0 and 6.0), as well as the SC1 () and SC2 modes ()

ob-tained for higher flame-interaction index (ni¼ 7:0 and 8.04at the

zero frequency limit). Three points in the SC2 trajectory with ni¼ 8:0 corresponding to several time delays (

s

=

s

0c ¼ 0 (A),

s

=

s

0

c ¼ 0:54 (B), and

s

=

s

0c¼ 0:9 (C)) are displayed inFig. 17and will

be used as typical cases to show pressure structures in the annular plenum and chamber.

For low flame interaction index (ni¼ 5:0 and 6.0), the first two

modes always have different frequencies and can be identified as plenum (WCPp: ) or chamber modes (WCCp: ), as studied in Section5.

However, for higher flame interaction indices (ni¼ 7:0 and 8.0),

a bifurcation occurs: frequencies of the annular plenum, burners, and annular chamber can match, leading to strongly coupled modes where the whole system resonates. The trajectory of the first strongly coupled mode (SC1: ) goes from the WCP1 mode (for small time delays

s

<

s

0

p=2) to a longitudinal mode (not presented here,

around 40 Hz, for large time delays

s

>

s

0

p=2). A second strongly

coupled mode (SC2: ) has a trajectory in the complex plane coming from the WCC1 mode (for small time delays

s

<

s

0

c=2) and going to

the WCP1 mode (for large time delays

s

>

s

0

c=2).

The stability map of the SC2 mode () at ni¼ 8:0 has been

val-idated against the 3D finite element solver AVSP (Fig. 18). Good agreement between AVSP () and the numerical resolution of Eq.

(25)( ) is found. The growth rate is slightly underestimated, but the global trend is well captured: the mode is fully unstable for all time delays, which corresponds to a new behavior compared to the weakly coupled modes. The analytical approach developed in Section5for the weakly coupled regime ( inFig. 18) is not able to capture this new highly nonlinear behavior.

When the flame interaction index is sufficiently large (ni> 7:0),

i.e., when the flame is sufficiently intense, the modes of plenum and chamber lock in and become unstable for all time delays.

Fig-ure 19shows the modulus of the coupling factors kC1k; kC4k and

the interaction product kC2C3k (terms appearing in analytical

re-sults summarized inTable B.1) as a function of the time delay

s

1

for the three different cases corresponding to the first order (p ¼ 1) chamber mode: n1¼ 1:57 (weakly coupled regime),

n1¼ 4:0 (limit case between weakly/strongly coupled regimes),

and n1¼ 8:0 (strongly coupled regime). It highlights the tuning

mechanism:

 In the weakly coupled regime (n1¼ 1:57), bothC1andC4have

the same order of magnitude and they vary in opposite direc-tions. The low-coupling-factors assumption is satisfied.  Then, at the limit case between weakly and strongly coupled

regimes (n1¼ 4:0), kC4k is almost constant with time delay

s

1.

The other coupling factors are not affected.

 Finally, in the strongly coupled regime (n1¼ 8:0), the coupling

parameter of the burner/chamber junction C4 is amplified,

which indicates that this junction is tuned. The evolution with the time delay is reversed compared to the weakly coupled regime: now, the two annular cavities (plenum and chamber) act together in the same manner; they are coupled. Conse-quently, the interaction term (kC2C3k) is also increased. The

low-coupling-factors assumption is not satisfied in this case. Pressure structures (p0_{¼ jp}0_{j cosðargðp}0_{ÞÞ) along the azimuthal}

direction in the plenum (PL line inFig. 9) and in the chamber (CL line inFig. 9) obtained with AVSP for the second strongly coupled mode are displayed inFig. 20for a high flame-interaction index (ni¼ 8:0) and several time delays (A:

s

=

s

0c¼ 0, B:

s

=

s

0c¼ 0:54,

and C:

s

=

s

0

c¼ 0:9 from Fig. 17). For null time delays (A in

Fig. 17), acoustic activity is only present in the chamber ( )

and the frequency is close to the weakly coupled chamber mode. The acoustic activity in the second annular cavity (i.e., the plenum: Fig. 16.Eigenfrequencies of both WCP1 and WCC1 modes (top) and zoom on WCP1

(bottom left) and WCC1 (bottom right) when the flame delay changes for PBC configuration with four burners (N ¼ 4) with ni¼ 0:2, 0.4, 0.6, and 0.8. : Passive

flame (ni¼ 0). : WCP1 mode. : WCC1 mode. I :s=s0p¼ 0 ors=s0c¼ 0

oriented in the increasingsdirection, . :s=s0

p¼ 1=2 ors=s0c¼ 1=2.

Fig. 17.Eigenfrequencies on the complex plane for a PBC configuration with four burners (N ¼ 4). : WCP1 mode (ni¼ 5:0 and 6.0). : WCC1 mode (ni¼ 5:0

and 6.0). : SC1 mode (ni¼ 7:0 and 8.0). : SC2 mode (ni¼ 7:0 and 8.0). I :s=s0p¼ 0

ors=s0

c¼ 0 oriented in the increasingsdirection and .:s=s0p¼ 1=2 ors=s0c¼ 1=2.

4_{Note that the typical order of magnitude of the interaction index n}

iis Tb=Tuÿ 1.

) grows with the time delay

s

. For

s

=

s

0

c¼ 0:54 ms (B inFig. 17),

a strong interaction with the burners appears leading to higher growth rates. Surprisingly, this case corresponds only to moderate acoustic activity in the second annular cavity (i.e., the annular ple-num: ). For

s

=

s

0

c¼ 0:9 (C inFig. 17), a first-order mode is

pres-ent in both annular cavities highlighting a strongly coupled situation. This strong interaction between plenum and chamber re-vealed by the presence of acoustic activity in both annular cavities leads, however, to a marginally unstable mode.

Figure 21shows the pressure phase in the annular plenum ( )

and in the annular chamber ( ) for several time delays. The same mode with opposite direction is also found with AVSP but not shown here. For small time delay (

s

=

s

0

c¼ 0 for case A and

s

=

s

0

c¼ 0:54 for case B ofFig. 17), pressures in the plenum and in

the chamber have different natures: purely spinning mode in the chamber (linear phase) and a mixed mode in the plenum (wave shape of the phase). The combination of clockwise and counter clockwise mode can generate purely spinning or purely standing mode but not necessarily in both annular cavities at the same time. However, higher time delay cases (such as case C, where

s

=

s

0

c¼ 0:9)

correspond to strongly coupled situations where both annular cav-ities lock in (Fig. 20C). In this case, pressure in the chamber and ple-num exhibit the same nature: purely spinning in both cavities in

Fig. 21C (the same mode with the opposite rotating direction is also

found by AVSP but not shown here). The combination of the purely spinning modes (clockwise and counter clockise) can also lead to standing modes in both annular cavities at the same time. This is a specific behavior only encountered in locked-in modes as case C.

Finally, it is demonstrated that a highly unstable mode does not necessarily exhibit strong acoustic activity in both annular cavities (as for case B) and that a mode where acoustic activity appears in the whole system can be only marginally unstable (as for case C). Moreover, the phase lag between pressure in the annular cavity and in the annular chamber as well as the nature of the mode (spinning, standing, or mixed), changes when acoustic activity is present in both annular cavities, a property that could be checked experimentally.

7. Conclusion

To complement expensive large eddy simulation [44] and Helmholtz [38] 3D codes used to study azimuthal modes in annular combustors, simple tools are required to understand the physics of these modes and create predesigns of industrial com-bustors. This paper describes a simple analytical theory to compute the azimuthal modes appearing in these combustors. It is based on a quasi-one-dimensional acoustic network where N burners are connected to an annular plenum and chamber. A manipulation of the corresponding acoustic equations in this configuration leads to a simple analytical dispersion relation that can be solved numer-ically. This method makes it possible to exhibit coupling factors be-tween plenum, burners, and chamber that depend on area ratios and flame transfer function (FTF). For N ¼ 4, a fully analytical res-olution can be performed in specific situations where coupling fac-tors of the FTF[22,26]are small and simple stability criteria can be Fig. 18.Eigenfrequency of the first-order chamber mode (p ¼ 1) for four burners (N ¼ 4) as a function ofs=s0

cwith ni¼ 8:0. : Numerical resolution of Eq.(25). :

Analytical model prediction for weakly coupled situations (Eq.(47)). : AVSP results.

Fig. 19.Modulus of the coupling factors kC1k (), kC4k (), and kC2C3k (þ) varying with the time delays1for three cases: n1¼ 1:57 (weakly coupled regime), n1¼ 4:0

proposed. For higher coupling factors, a bifurcation occurs, yielding a strongly coupled regime where acoustic activity is present in both annular cavities. The nature of such a mode (standing, spin-ning, or mixed) changes with the time delay. Purely spinning or purely standing modes in the annular plenum and in the chamber were found simultaneously only when a strongly coupled situation occurred (Fig. 21C). For strongly coupled cases, a PBC configuration where two annular cavities are connected to N burners is required to predict correctly eigenmodes and stability maps. This analytical tool has been compared systematically to the predictions of a full three-dimensional Helmholtz solver. A very good agreement is found, showing that the present asymptotic resolution is correct.

These results show that a simple analytical formulation for study-ing azimuthal modes in annular combustors is an intereststudy-ing tool that can be used as a predesign tool or as a help to postprocess acoustic or LES simulations.

Appendix A. Analytical dispersion relation of a PBC configuration with a single burner (N ¼ 1)

The analytical dispersion relation (Eq.(25)) is obtained for a general PBC configuration with N burners. To explain the ATACA-MAC approach leading to the analytical eigenfrequencies of the system, the case of a single burner (N ¼ 1) will be detailed.5

Considering only one burner, the transfer matrix M ¼ R1T1ÿ Id

of the whole system is Fig. 20.Pressure structures and isolines (p0_{¼ jp}0_{j cosðargðp}0_{ÞÞ) obtained with AVSP}

(left) and pressure along the azimuthal direction in the annular chamber ( ) and annular plenum ( ) for the SC2 mode of a PBC configuration with four burners (N ¼ 4) and ni¼ 8:0 at several time delays:s=s0c¼ 0 (top), 0.54 (middle),

and 0.9 (bottom). The configuration corresponds to fFDP1¼ 56 Hz and fFDC1¼ 90 Hz.

Fig. 21.Pressure phase obtained with AVSP for the SC2 mode at ni¼ 8:0 along the

azimuthal direction in the annular chamber ( ) and the annular plenum ( ) for several time delays:s=s0

c¼ 0 (top), 0.54 (middle), and 0.9 (bottom).

M ¼

cosð2kuLpÞ ÿ 1 ÿ sinð2kuLpÞ

sinð2kuLpÞ þ 2 cosð2kuLpÞ

C

1 ÿ2 sinð2kuLpÞ

C

1þ cosð2kuLpÞ ÿ 1

0 0 2 cosðkuLpÞ

C

3 ÿ2 sinð2kuLpÞ

C

3


0 0 2 cosð2kLcÞ

C

2 ÿ2 sinð2kLcÞ

C

2 cosð2kLcÞ ÿ 1 ÿ sinð2kLcÞ

sinð2kLcÞ þ 2 cosð2kLcÞ

C

4 ÿ2 sinð2kLcÞ

C

4þ cosð2kLcÞ ÿ 1 2 6 6 6 4 3 7 7 7 5: ðA:1Þ

5 _{Since there is only one burner, the index i has been omitted to simplify notation}

In the general case with N burners, a Taylor expansion of this matrix has to be performed as a first simplification. Here, the ma-trix is simple enough to compute analytically the determinant leading to the exact dispersion relation

ð

C

1

C

4ÿ

C

2

C

3Þ sinð2kLcÞ sinð2kuLpÞ þ 2

C

1½1 ÿ cosð2kLcފ sinð2kuLpÞ þ 2

C

4½1 ÿ cosð2kuLpފ sinð2kLcÞ

þ 4½1 ÿ cosð2kLcފ½1 ÿ cosð2kuLpފ ¼ 0: ðA:2Þ

The dispersion relation, Eq.(A.2), is nonlinear. The idea is to use a Taylor expansion at the second (or third) order and to solve it analytically. The expansion has to be done around a FDCp (i.e., kLc¼ p

p

þ



c) or FDPp (i.e., kuLp¼ p

p

þ



p) mode (see Section3.3

for details). For instance, in the case of the WCC1 mode (kLc¼

p

þ



, which implies kuLp¼ bð

p

þ



cÞ, where b ¼c

0_L p

c0 uLc), the

dispersion relation Eq.(A.2)becomes

½cosð2

p

bÞ ÿ 1Š



c

C

0₄þ



2cþ o



2c ÿ

h i

¼ 0; ðA:3Þ

whereC04is the value ofC4evaluated at kLc¼ p

p

.

Note that solutions of Eq.(A.3)being



c¼ ÿC04, the term



cC04is

of the same order of magnitude as



2

cand therefore has to be kept

in the analytical dispersion relation, Eq.(A.2).

Analytical dispersion relations for N > 1 are more complex to derive but follow a similar procedure. When N ¼ 4, the dispersion relations ofTable A.1are obtained.

Appendix B. Stability criterion of weakly coupled modes for a four-burner configuration (N ¼ 4)

A mode is stable if the imaginary part of the wavenumber is negative.Table B.1shows analytical expressions for the wavenum-ber perturbation



for WCPp and WCCp modes.6

Analytical stability criteria can be derived by calculating the sign of Im C01   ; Im C04   , and Im C02C 0 3  

using the following defini-tions: F _{is the complex conjugate of the flame parameter F ¼}

q0_c0 q0 uc0u 1 þ n:e jx0_s   , h0 ¼

x

0_{ð1 ÿ}

_a

_ÞL i=c02 R, and h0u¼

x

0

a

Li=c0u2 R.

The notation D refers to D ¼ j cosðh0

Þ sinðh0uÞ þ F sinðh 0

Þ cos h0u

ÿ
j2. With these notations, the signs of the imaginary parts of these coupling parameters are

Im

C

01   ¼ Si 4SpDsinð2h 0 ÞImðFÞ ðB:1Þ Im

C

0₄¼ ÿ Si 4ScDsin 2h 0 u ÿ
_ImðFÞ: ðB:2Þ

Eqs.(B.1) and (B.2)lead to simple analytical stability criteria for

WCCp and WCPp modes: sin 2

ps

=

s

0 c ÿ
_{sin 2p}

p

a

Lic0 Lcc0 u   < 0 for WCCp modes ðB:3Þ sin 2

ps

=

s

0 p   sin 2p

p

ð1 ÿ

a

ÞLic0u Lpc0   > 0 for WCPp modes: ðB:4Þ

Appendix C. Flame position effect on annular combustor stability

Similarly to longitudinal modes in the Rijke tube[2,26,43], the flame position (defined by

a

) also controls the stability (Eq.(48)). In a quasi-isothermal Rijke tube, for common (small) values of the FTF time delay

s

, stability of the first longitudinal mode is ob-tained only when the flame is located in the upper half of the tubes

[45,46], i.e.,

a

> 1=2, which can be extended for the pth

longitudi-nal mode:

2m þ 1 2p <

a

<

2ðm þ 1Þ

2p ;

8

m 2 N ðRijke tubeÞ: ðC:1Þ

Eq.(48)highlights a similar behavior for azimuthal modes in a PBC configuration: for a WCCp mode with small values of the time delay

s

<

s

0

c=2; sin 2

p

ss0 c

 

is positive and Eq.(48)leads to

2m þ 1 2p Lcc0 u Lic0<

a

< 2ðm þ 1Þ 2p Lcc0 u Lic0;

8

m 2 N ðWCCp modesÞ: ðC:2Þ

Usually, the critical flame position

a

crit¼ Lcc

0 u

2pLic0 is larger than

unity because the half-perimeter of the annular cavity is much longer than the burner length (Lc Li). Since the range of the

normalized flame position

a

is [0–1], the flame position may affect the stability only for high-order modes (i.e., p large enough to get

a

crit< 1). For instance, in the case described inTable 1with

the corrected burner length Li’ 0:76 m, the critical flame

posi-tions

a

crit and the stability ranges (Eq. (C.2)) are shown in Table C.1.

The change of stability with the flame position

a

for small time delays predicted inTable C.1has been validated using the numer-ical resolution of the dispersion relation (Eq.(25)) inFig. C.1. The critical flame positions obtained in Table C.1 are well captured for all modes. A situation where the plenum/chamber interaction is not negligible is shown for the WCC1 mode with

a

¼ 0:3 (i.e., the flame is close to the pressure node imposed by the large annu-lar plenum).

Table A.1

Analytical expressions of wavenumber perturbation for WCCp and WCPp azimuthal modes.

Type Odd/even Second-order dispersion relation (oð2_Þ)

WCC Odd _sinðppbÞ2_{þ 4C}0 4þ 4C04 2 h i ¼ 0 Even _{sinðppb=2Þ}2_{þ 4C}0 4 h i ¼ 0 WCP Odd _sinðpp=bÞ2_{þ 4C}0 1þ 4C01 2 h i ¼ 0 Even _{sinðpp=ð2bÞÞ}2_{þ 4C}0 1 h i ¼ 0 Table B.1

Analytical expressions for wavenumber perturbationfor WCCp and WCPp modes where HðxÞ ¼ 4 tanðppx=2Þ and GðxÞ ¼ 4 sinðppx=2Þ

cosðppx=2Þÿðÿ1Þp=2have real values. Type Odd/even Wavenumber perturbation ()

WCC Odd ÿ2C0 4ÿ HðbÞC02C03 Even ÿ2C0 4ÿ GðbÞC02C03 WCP Odd ÿ2C0 1ÿ Hð1=bÞC02C03 Even ÿ2C0 1ÿ Gð1=bÞC02C03 Table C.1

Critical flame positionsacritand flame positions satisfying Eq.(C.2)for WCCp

odd-order modes of the case described inTable 1.

Mode order (p) p ¼ 1 p ¼ 3 p ¼ 5 p ¼ 7 acrit¼ Lcc 0 u 2pLic0 2.70 0.9 0.54 0.39 aSatisfying Eq.(C.2) none [0.9–1] [0.54–1] [0.39–0.78]

6 _{Since all sectors are identical, the index i has been omitted to simplify notations}